You just misunderstood how this map works. Sources of his income are:
- 4 armies/turn - base income;
- 1 army/turn - Istanbul bonus;
- 1 army/turn - Mecca bonus;
- 15 armies/turn - everything else.
Why 15? Because he has 6 of those "strategic points". Map says that for every two of them you get +1 army/turn, and that's true. But that, in fact, does not mean "you get N/2 armies per turn, where N is number of strategic points under your control". It means "for every set of two strategic points both of which are under your control, you get 1 army/turn". So, if you label his territories as A, B, C, D, E and F, he has 15 such sets: (A&B), (A&C), (A&D), (A&E), (A&F), (B&C), (B&D), (B&E), (B&F), (C&D), (C&E), (C&F), (D&E), (D&F) and (E&F). If you want easy formula, it's (N*(N-1))/2
, where N is number of strategic points under payer's control.